The Symmetries of Things

John H. Conway, Heidi Burgiel + Chaim Goodman-Strauss

The Symmetries of Things begins with repeating plane patterns and
simple and intuitive ideas of reflection and rotation, but finishes with
four and higher dimensional symmetry groups, offering an extraordinary
illustration of just how rapidly mathematics can climb in abstraction
and difficulty from elementary foundations.

Part I has some substantial ideas in it, but assumes "no particular
mathematical skills" and could be made accessible to primary school
children. Part II is pitched at undergraduate level, as it assumes a
solid familiarity with group theory. And Part III includes some novel
research results: Conway et al. expect it "will be completely understood
only by a few professional mathematicians".

Part I, "Symmetries of Finite Objects and Plane Repeating Patterns",
classifies the seventeen plane repeating patterns — sometimes referred
to as "wallpaper" patterns, though few of them have ever been so used —
using a new signature notation, and continues with the fourteen spherical
patterns (including seven infinite families) and the seven frieze types.
The approach here is very much visual, built around kaleidoscopes,
gyrations and so forth and without any formal apparatus of group
theory, affine transformations, or suchlike. Indeed it works backwards,
proving the completeness of the classifications using "Magic Theorems",
justifying those using Euler's Theorem V - E + F = 2, and only then
proving that Theorem. It concludes with the classification of surfaces
and the concept of an orbifold.

Some of this seems like it is actually aimed at children: the Magic
Theorems are presented using "dollars", the proof of Euler's Theorem
is turned into a game with barbarian sea-raiders breaking dykes and
sacking towns, and there are some simple quizzes (identifying patterns
from visualizations) and brief "where are we?" summaries at the end of
chapters. And I can't see why it wouldn't be usable in primary schools —
I had to restrain myself from trying to explain it to my three year old —
where it would make a refreshing change from times tables and arithmetic.

Part II, "Color Symmetry, Group Theory, and Tilings", goes on to look at
twofold and threefold and primefold colorings of patterns on the plane
and sphere, and at isohedral tilings. A bit of group theory is used
here — in particular the idea of a group presentation — and there
is some exploration of abstract groups, focusing on the group number
function gnu(n), the number of distinct abstract groups of order n.
There's a brief attempt to explain some of the theory required, but this
material really presupposes a first course in group theory.

Part III, "Repeating Patterns in Other Spaces", considers repeating
patterns in the hyperbolic plane, Archimedean and Catalan polyhedra and
tilings, generalised Schläfli symbols (for representing all topologically
spherical polyhedra), the 35 "prime" space groups and the matching
objects, the 184 composite space groups ("that preserve at least one
family of parallel lines, and so in a sense are composed of one- and
two-dimensional groups, which makes them less interesting") and the flat
(Euclidean) three-dimensional spaces (ten compact and eight infinite),
before glancing into the fourth and higher dimensions.

As warned, a good bit of this did go over my head, but I followed enough
of it to appreciate what was being done.

"We marry a regular polyhedron P and its dual Q by placing
them so that corresponding edges intersect at right angles. Then,
their daughter polyhedron is the Archimedean polyhedron that
is their intersection, whose dual, their son, is the Catalan
one that is their convex hull."

There are also many attractive illustrations here, though some of these
— such as the attempts to convey the structure of 3-manifolds — are
not that easy to wrap one's head around.

More generally, The Symmetries of Things is full of attractive colour
illustrations, mostly more accessible, of patterns, orbifolds, group
presentations, tilings, polyhedra and so forth. Created for this book
and not, with the exception of a few Escher figures, taken from art
or architecture, these are essential to understanding the material.
When introducing orbifolds, for example, the authors write: "There is
very little text: we prefer to explain things largely by picture."

One major drawback of The Symmetries of Things is its cost: it is
priced as a research monograph rather than a popular work or a general
textbook, and seems expensive even given the high quality binding and
paper and colour. Some of it at least deserves quite a broad audience;
my suggestion would be to republish Part I separately as a paperback,
perhaps even augmented with materials for school use.